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Journal of the Physical Society of Japan

Time Reversal Invariant Evolution Equation in Dilute Quantum Gas

Shokichi Kanno

Published Jul 15, 2001• 0 citations
Publisher

Abstract

For a dilute gas interacting through a short range potential, we have obtained, by starting from the quantum mechanical Hamiltonian, a closed set of evolution equations for the particle distribution and for the expectation values of the four-field-operator products. The set of equations is invariant under time reversal. The evolution equation for the particle distribution depends on the initial conditions, and reduces to the Boltzmann transport equation for almost all initial states.

DOI
10.1143/jpsj.70.1931

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For a dilute gas interacting through a short range potential, we have obtained, by starting from the quantum mechanical Hamiltonian, a closed set of evolution equations for the particle distribution and for the expectation values of the four-field-operator products. The set of equations is invariant under time reversal. The evolution equation for the particle distribution depends on the initial conditions, and reduces to the Boltzmann transport equation for almost all initial states.
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We explore whether quantum field theory can be understood as the statistical mechanics of a time-reversal-invariant stochastic generalization of Hamiltonian dynamics. The motivation for this project, started with this paper, is to assign sharp values to all observables and thereby avoid the quantum measurement problem. In classical mechanics, motion is deterministic and corresponds to an evolution of the phase space probability density according to Liouville's equation that is governed by first derivatives of the Hamiltonian in phase space. We derive a generalization of the Liouville equation with natural constraints -- namely, reduction to classical Hamiltonian dynamics as the stochasticity parameter ℏ↦0\hbar\mapsto0ℏ↦0, Fokker-Planck form for the probability density evolution, local Hamiltonian dependence, time-reversal invariance, energy conservation, and minimality -- which turns out to be a Fokker-Planck equation with a generalized diffusion matrix that is symmetric, traceless, and constructed from the Hessian of the Hamiltonian. We then show that the Schrödinger equation in the coherent-state phase-space formulation of certain bosonic QFTs has precisely this form, with the Husimi function playing the role of the phase space probability density. The question to what extent this equation can be interpreted in terms of objective stochastic field theories is discussed in a companion paper.
Time‐reversal symmetry is a fundamental property of many quantum mechanical systems. The relation between statistical physics and time reversal is subtle, and not all statistical theories conserve this particular symmetry—most notably, hydrodynamic equations and kinetic equations such as the Boltzmann equation. Here, we consider quantum kinetic generalizations of the Boltzmann equation using the method of reduced density operators, leading to the quantum generalization of the Bogolyubov‐Born‐Green‐Kirkwood‐Yvon (BBGKY) hierarchy. We demonstrate that all commonly used approximations, including Vlasov; Hartree‐Fock; and the non‐Markovian generalizations of the Landau, T‐matrix, and Lenard‐Balescu equations, are originally time‐reversal invariant, and we formulate a general criterion for time reversibility of approximations to the quantum BBGKY hierarchy. Finally, we illustrate, through the example of the Born approximation, how irreversibility is introduced into quantum kinetic theory via the Markov limit, making the connection with the standard Boltzmann equation. This paper is a complement to paper I (Scharnke et al., J. Math. Phys., 2017, 58, 061903), where the time‐reversal invariance of quantum kinetic equations was analysed in the frame of the independent non‐equilibrium Green functions formalism.
Basing his work on a new formulation of thermodynamics called the Unified Quantum Theory of Mechanics and Thermodynamics first published in a series of four ground breaking papers in 1976 (Hatsopoulos and Gyftopoulos, 1976a, b, c, d), Beretta develops a dynamical postulate (Beretta et al. 1984; Beretta, Gyftopoulos, and Park, 1985) consistent both with the non-dynamical quantum mechanical postulates of the Unified Theory as well as with its thermodynamic ones (the 2nd Law in particular). The theory itself simply and elegantly extends in a unified fashion the concepts of thermodynamics to quantum mechanics and the concepts of quantum mechanics to thermodynamics. It does so without the bridge traditionally used, i.e. statistical mechanics, eliminating a number of the ambiguities, tautologies, and inconsistencies (including a built-in violation of the 2nd Law) inherent in the presentations of both Classical and Statistical Thermodynamics. This new formulation generalizes thermodynamics so that it applies to all systems large or small (including one particle systems) either in a state of thermodynamic (i.e. stable) equilibrium or not in a state of thermodynamic equilibrium. The Beretta equation of motion describes the time evolution of the state of a system via a density operator which is uniquely based on an unambiguous preparation of an ensemble of identical systems, i.e. the so-called homogenous or irreducible ensemble, and does so both for unitary and non-unitary reversible as well as irreversible processes. In this paper, we present a simple application of this general equation of motion to the time evolution of the entropy of a closed system comprised of a Boltzmann type gas consisting of one or of many particles undergoing an irreversible process. A number of different energy eigenlevels and initial states and their effects on entropy generation and the final state of maximum entropy, i.e. stable equilibrium, are examined. A simple time-dependent work interaction is introduced into the formulation to show how this in turn affects the evolution of the state of the system.